Abstract

We consider the Hyers-Ulam stability for the following fractional differential equations in sense of Srivastava-Owa fractional operators (derivative and integral) defined in the unit disk: , in a complex Banach space. Furthermore, a generalization of the admissible functions in complex Banach spaces is imposed, and applications are illustrated.

1. Introduction

A classical problem in the theory of functional equations is the following: if a function approximately satisfies functional equation , when does there exist an exact solution of which approximates? In 1940, Ulam [1, 2] imposed the question of the stability of Cauchy equation, and in 1941, Hyers solved it [3]. In 1978, Rassias [4] provided a generalization of Hyers theorem by proving the existence of unique linear mappings near approximate additive mappings. The problem has been considered for many different types of spaces (see [5–7]). Li and Hua [8] discussed and proved the Hyers-Ulam stability of spacial type of finite polynomial equation, and Bidkham et al. [9] introduced the Hyers-Ulam stability of generalized finite polynomial equation. Rassias [10] imposed a Cauchy type additive functional equation and investigated the generalized Hyers-Ulam “product-sum” stability of this equation.

Recently, Jung presented a book [11], which complements the books of Hyers, Isac, and Rassias (Stability of Functional Equations in Several Variables, Birkhäuser, 1998) and of Czerwik (Functional Equations and Inequalities in Several Variables, World Scientific, 2002) by covering and offering almost all classical results on the Hyers-Ulam-Rassias stability such as the Hyers-Ulam-Rassias stability of the additive Cauchy equation, generalized additive functional equations, Hosszú’s functional equation, Hosszú’s equation of Pexider type, homogeneous functional equation, Jensen’s functional equation, the quadratic functional equations, the exponential functional equations, Wigner equation, Fibonacci functional equation, the gamma functional equation, and the multiplicative functional equations. Furthermore, the concept of superstability for some problems is defined and studied.

The Ulam stability and data dependence for fractional differential equations in sense of Caputo derivative has been posed by Wang et al. [12] while in sense of Riemann-Liouville derivative has been discussed by Ibrahim [13]. Finally, the author generalized the Ulam-Hyers stability for fractional differential equation including infinite power series [14, 15].

The class of fractional differential equations of various types plays important roles and tools not only in mathematics but also in physics, control systems, dynamical systems and engineering to create the mathematical modeling of many physical phenomena. Naturally, such equations required to be solved. Many studies on fractional calculus and fractional differential equations, involving different operators such as Riemann-Liouville operators [16], Erdèlyi-Kober operators [17], Weyl-Riesz operators [18], Grünwald-Letnikov operators [19] and Caputo fractional derivative [20–24], have appeared during the past three decades. The existence of positive solution and multipositive solutions for nonlinear fractional differential equation are established and studied [25]. Moreover, by using the concepts of the subordination and superordination of analytic functions, the existence of analytic solutions for fractional differential equations in complex domain is suggested and posed in [26–28].

2. Preliminaries

Let be the open unit disk in the complex plane and denote the space of all analytic functions on . Here we suppose that as a topological vector space endowed with the topology of uniform convergence over compact subsets of . Also for and , let be the subspace of consisting of functions of the form
Let be the class of functions , analytic in and normalized by the conditions . A function is called univalent if it is one-one in . A function is called convex if it satisfies the following inequality:
We denoted this class .

In [29], Srivastava and Owa, posed definitions for fractional operators (derivative and integral) in the complex -plane as follows.

Definition 2.1. The fractional derivative of order is defined, for a function by
where the function is analytic in simply connected region of the complex -plane containing the origin, and the multiplicity of is removed by requiring to be real when .

Definition 2.2. The fractional integral of order is defined, for a function , by
where the function is analytic in simply connected region of the complex -plane containing the origin, and the multiplicity of is removed by requiring to be real when .

Remark 2.3. We have the following:
In [27], it was shown the relation
More details on fractional derivatives and their properties and applications can be found in [30, 31].We next introduce the generalized Hyers-Ulam stability depending on the properties of the fractional operators.

Definition 2.4. Let . We say that
has the generalized Hyers-Ulam stability if there exists a constant with the following property: for every , , if
then there exists some that satisfies (2.7) such that
In the present paper, we study the generalized Hyers-Ulam stability for holomorphic solutions of the fractional differential equation in complex Banach spaces and
where and are holomorphic functions such that ( is the zero vector in ).

3. Generalized Hyers-Ulam Stability

In this section we present extensions of the generalized Hyers-Ulam stability to holomorphic vector-valued functions. Let represent complex Banach space. The class of admissible functions consists of those functions that satisfy the admissibility conditions:
We need the following results.

Lemma 3.1 (see [32]). If is holomorphic, then is a subharmonic of . It follows that can have no maximum in unless is of constant value throughout .

Lemma 3.2 (see [33]). Let be the holomorphic vector-valued function defined in the unit disk with (the zero element of ). If there exists a such that
then

Lemma 3.4 (see [29, page 225]). If the function is in the class , then

Theorem 3.5. Let and be a holomorphic vector-valued function defined in the unit disk , with . If , then

Proof. Since , then from Lemma 3.3, we observe that
Assume that for . Thus, there exists a point for which . According to Lemma 3.1, we have
In view of Lemma 3.2, at the point , there is a constant such that
Consequently, we obtain that
We put , for some and and ; hence from (3.1), we deduce
which contradicts the hypothesis in (3.6) that we must have .

Corollary 3.6. Assume the problem (2.10). If is a holomorphic univalent vector-valued function defined in the unit disk , then

Proof. By univalency of , the fractional differential equation (2.10) has at least one holomorphic univalent solution . Thus, according to Remark 2.3, the solution of the problem (2.10) takes the form
Therefore, in virtue of Theorem 3.5, we obtain the assertion (3.12).

Theorem 3.7. Let be holomorphic univalent vector-valued functions defined in the unit disk then (2.10) has the generalized Hyers-Ulam stability for .

Proof. Assume that
therefore, by Remark 2.3, we have
Also, and thus . According to Theorem 3.5, we have
Let and be such that
We will show that there exists a constant independent of such that
and satisfies (2.7). We put the function
thus, for , we obtain
Without loss of generality, we consider yielding
This completes the proof.

In the same manner of Theorem 3.5, and by using Lemma 3.4, we have the following result.

Theorem 3.8. Let and be a holomorphic vector-valued function defined in the unit disk , with . If , then

4. Applications

In this section, we introduce some applications of functions to achieve the generalized Hyers-Ulam stability.

Example 4.1. Consider the function by
with , and . Our aim is to apply Theorem 3.5, this follows since
when , . Hence by Theorem 3.5, we have the following. If , and is a holomorphic univalent vector-valued function defined in , with , then
Consequently, , thus in view of Theorem 3.7, has the generalized Hyers-Ulam stability.

Example 4.2. Assume that the function by
with . By applying Corollary 3.6, we need to show that . Since
when , and . Hence by Corollary 3.6, we have the following. For is a holomorphic vector-valued function defined in , with , then
Consequently, , thus in view of Theorem 3.7, has the generalized Hyers-Ulam stability.

Example 4.3. Let satisfy the following:
for every , and . Consider the function by
with . Now for , we have
and thus . If is a holomorphic vector-valued function defined in , with , then
Hence according to Theorem 3.7, has the generalized Hyers-Ulam stability.

Acknowledgment

This research has been funded by University Malaya, under Grant no. (RG208-11AFR).